Hs
1
s À s
1
s À s
2
s À s
3
1
s À e
j2p=3
s À e
jp
s À e
j4p=3
1
s 1s
2
s 1
:
Chebyshev filters permit a certain amount of ripples in the passband, but have a much
steeper roll-off near the cut-off frequency than what the Butterworth design can achieve.
The Chebyshev filter is called the equiripple filter because the ripples are always of equal
size throughout the passband. Even if we place very tight limits on the passband ripple,
the improvement in roll-off is considerable when compared with the Butterworth filter.
There are two types of Chebyshev filters. Type I Chebyshev filters are all-pole filters
that exhibit equiripple behavior in the passband and a monotonic characteristic in the
stopband (see Figure 6.7a). The family of type II Chebyshev filters contains both poles
and zeros, and exhibit a monotonic behavior in the passband and an equiripple behav-
ior in the stopband, as shown in Figure 6.7(b). In general, the Chebyshev filter meets the
specifications with a fewer number of poles than the corresponding Butterworth filter.
Although the Chebyshev filter is an improvement over the Butterworth filter with
respect to the roll-off, it has a poorer phase response.
The sharpest transition from passband to stopband for any given d
p
, d
s
, and L can be
achieved using the elliptic design. In fact, the elliptic filter is the optimum design in this
sense. As shown in Figure 6.8, elliptic filters exhibit equiripple behavior in both the
H(s)
H(
-s)
s
2
s
1
s
0
s
5
s
4
s
3
j
s
Figure 6.6 Poles of the Butterworth polynomial for L 3
(b)
(a)
1
- d
p
|H(
)|
1
d
s
p
s
1
- d
p
|H(
)|
1
d
s
p
s
Figure 6.7 Magnitude responses of Chebyshev lowpass filters: (a) type I, and (b) type II
252
DESIGN AND IMPLEMENTATION OF IIR FILTERS